Distinct Distances Between a Circle and a Generic Set
Alex McDonald, Brian McDonald, Jonathan Passant, Anurag Sahay
aa r X i v : . [ m a t h . M G ] M a y Distinct Distances from Points on a Circle to a Generic Set
Alex McDonald, Brian McDonald, Jonathan Passant and Anurag SahayMay 7, 2020
Abstract
Let S be a set of points in R contained in a circle and P an unrestricted point set in R . We prove thenumber of distinct distances between points in S and points in P is the minimum of | S | | P | , | P | and | S | . Thisbuilds on work of Pach and De Zeeuw [15], Bruner and Sharir [3], McLaughlin and Omar [13] and Mathialagan[12] on distances between pairs of sets. In 1945 Erd˝os introduced his distinct-distances problem, first stated in [7], asking for the minimum number ofdistinct distances an n point set can create in R . Erd˝os showed that a square lattice Λ of n points determined | ∆ p Λ q| À n ? log n distances, where here and throughout, Á and À are used to suppress some constant independentof the controlling parameter and ∆ p S q denotes the set of distances between elements of S . Erd˝os conjectured thatthis was essentially the best possible i.e. for any n point set P one has | ∆ p P q| Á n ´ ε , for all ε ą
0. This questionplayed a key role in the combinatorial geometry for over 50 years, with many successive improvements, see [2,Section 5.3] or [8]. In [9] Guth and Katz provided a solution utilising significant new algebraic developments inwhat has become known as the polynomial method in combinatorics.A natural variant of the distinct distances problem asks for the minimum number of distances between points a P A, b P B where one or both of the finite sets A and B are constrained in some way. Purdy [2, Section 5.5]considers a version of this problem where ℓ and ℓ are lines in the plane, and A and B consist of n points on ℓ and ℓ , respectively. He observed that if ℓ and ℓ are parallel or perpendicular then one may imitate the grid examplegiven by Erd˝os to obtain À n distances, but conjectured that otherwise the number of distnaces was superlinear.More precisely, he conjectured that for every C there exists n such that if A Ă ℓ , B Ă ℓ each have size n ą n and determine ă Cn distances, then ℓ and ℓ are either parallel or perpendicular. Elekes and R´onyai [6] provethis conjecture from a statement about restricted polynomials, implicitly showing that there exists δ ą p A, B q Á n ` δ under the conditions of the conjecture. Elekes [5] subsequently showed that one can in fact take δ “ . Schwartz, Solymosi, and de Zeeuw prove that in the unbalanced version of the problem, where | A | “ n ` ε and | B | “ n , the number of distances is still superlinear. The results in both the balanced and unbalanced caseswere improved by Sharir, Sheffer and Solymosi [19] who use algebraic techniques to show that sets A and B of size n and m , respectively, satisfying the hypotheses above, determine Á min t n { m { , n , m u distances.This question has been generalised in two key ways. Pach and de Zeeuw [15] showed that if you restrict your pointsets to two algebraic curves C and C of constant degree (constant with respect to n and m the number of pointson C and C respectively), then one has at least Á min t n { m { , n , m u distinct distances, provided the curvesare not parallel lines, orthogonal lines, or concentric circles. This argument used heavily that both point sets are oncurves, so their role could be interchanged and that curves that are not parallel lines, orthogonal lines, or concentriccircles don’t share too many symmetries.In a different direction Bruner and Sharir [3] showed that when the first set P of m points is on a line and thesecond P of n points is unrestricted in the plane one has at least | ∆ p P, P q| Á min " n { m { , m { n { log { n , n , m * . m points P to a curve of constant degree and have n unrestricted points P one has | ∆ p P, P q| Á m { n { log ´ { n when m Á n { log ´ { n,m { n { when m À n { log ´ { n. Finally, Mathialagan [12] extended these results in R to the setting where P and P are both unrestricted pointsets (of size m and n respectively) | ∆ p P, P q| Á m { n { log ´ n when n { ď m ď n,m { n { when m ď n { Since P and P are symmetric in this case, analogous bounds hold when m ě n . In particular, this subsumesMcLaughlin-Omar’s result.The aim of this paper is to find similar results in the case where the first set lies on a given circle and the secondset is essentially unrestricted. We obtain the following result. Theorem 1.
Suppose that S is a point set on a circle in R and P is a point set in R such that no two points of P are on any concentric circle. Then we have, | ∆ p S, P q| Á min t| S | | P | , | P | , | S | u . For comparison with the Theorems stated above we use | S | “ m and | P | “ n we have | ∆ p S, P q| Á min t m n , n , m u . Note that the the theorem would be false without the hypothesis that no two points in P lie on a circle about theorigin (though two could be replaced by another fixed constant). To see this, let S be a set of n points on the unitcircle, evenly spaced. Let P “ αS for some positive real number α . For a fixed p P P , | ∆ pt p u , S q| ď | S | “ n But by symmetry, ∆ pt p u , S q “ ∆ pt p u , S q for any p, p P P . Thus, | ∆ p P, S q| “ | ∆ pt p u , S q| ď n , which wouldcontradict the conclusion of the theorem.We note that Theorem 1 is probably far from sharp. We would be extremely interested in any example that hadfewer than c | S || P | distances (with points not on concentric circles).A note on the proof of Theorem 1 compared to the results mentioned above. We follow a similar approach to [5],[19], [15] and [3] in that we use the restriction of S to an algebraic curve to set up new ‘curves’ in R , where anincidence in these new curves with a pair of points from our circle set correspond to a repeated distance. Thuswe care about providing an incidence bound for these curves and points. This is where we use a new element inour proof, showing via a version of B´ezout (See Theorem 4) that no five points can contain 17 of these constructedcurves. This proof relies on the fact that the polynomials we use to form our curves are sufficiently independent,heavily using that our first set is restricted to the circle.Note that for Pach-de Zeeuw [15], assuming one of the curves is a circle, and the other isn’t a concentric circle,our result applies in a setting with one point set having no restrictions. However, our bound is much weaker; thesituation where our main term would be better than theirs corresponds to when S and P are unbalanced, with | S | being much larger than | P | (specifically | S | ě | P | ). In this regime, the | P | term is already dominant in both theirbound and ours.The relationship between Mathialagan [12] and our work is a little more complicated. Mathialagan’s results aremuch more general than ours, as neither of the sets is restricted at all. Thus, we can expect to improve these bounds2hen restricted to our setting. In the close to balanced cases (when | P | Æ | S | À | P | , where by Æ we mean thatthe inequalities hold up to log-factors in | S | or | P | ) we get a better lower bound. However, in the unbalanced cases,(when | S | Á | P | and | S | Æ | P | ) Mathialagan’s bounds are better than ours, despite our more restrictive setting.In Section 2 we provide the initial framework for the bound on | ∆ p S, P q| , repeating the idea of Elekes that one canuse pairs of repeated distances and we discuss the incidence bound we will use. In Section 3 we apply B´ezout tobound the number of curves through a specific set of points, setting up the necessary conditions for the incidencebound (see Corollary 1). Finally in Section 4 we restrict our curves to S ˆ S and prove a technical Lemma thatallows us to finally apply our incidence bound. The authors wish to thank Adam Sheffer for introducing us to the problem, pointing out the reference [12] andencouragement. The third author wishes to thank Adam Sheffer, Josh Zahl and the participants of the MSRI summerschool on the Polynomial method for many helpful discussions and MSRI, Berkeley for hosting the workshop.
Definition 1.
Given any two finite sets
S, P Ă R , define the distance set and quadruple set of S and P , respectively,as ∆ p S, P q “ t| u ´ p | : u P S, p P P u ,Q p S, P q “ tp u, v, p, q q P S ˆ P : | u ´ p | “ | v ´ q |u . Theorem 2.
For any finite sets of points
S, P Ă R , we have | ∆ p S, P q| ě | S | | P | | Q p S, P q| Proof.
The statement follows directly from the classic Cauchy-Schwarz energy bound. Let v p t q “ tp u, p q P S ˆ P : | u ´ p | “ t u be the number of occurrences of the distance t , then we have | S | | P | “ ¨˝ ÿ t P ∆ p S,P q v p t q ˛‚ ď | ∆ p S, P q| ÿ t v p t q “ | ∆ p S, P q| ¨ | Q p S, P q| . Therefore, an upper bound on the size of Q p S, P q will yield a lower bound on the size of ∆ p S, P q . In order to bound Q p S, P q we will follow the approach of Pach and de Zeeuw [15] and Bruner and Sharir [3], setting up an incidenceproblem. We start by making a couple simple reductions. First, without loss of generality we may assume the circle S lives on is the unit circle centered at the origin, so all u P S satisfy } u } “
1. Second, note that for any point p P P and distance t , there are at most two choices of u P S for which } p ´ u } “ t , since the circle centered at p of radius t can only intersect the unit circle twice. Therefore, the number of quadruples p u, v, p, q q P Q p S, P q with p “ q is « | S || P | . It remains to bound our modified quadruple set r Q p S, P q “ tp u, v, p, q q P S ˆ P : p ‰ q, } p ´ u } “ } q ´ v }u . Let Π “ S and let Γ “ t Z R p f p,q q : p, q P P, p ‰ q u , where3 p,q p x, x , y, y q “ p p ´ x q ` p p ´ x q ´ p q ´ y q ´ p q ´ y q , and Z R p f q refers to the points p x, x , y, y q in R such that f p x, x , y, y q “
0. If u “ p x, x q , v “ p y, y q , then p u, v, p, q q P r Q p S, P q if and only if p ‰ q and f p,q p x, x , y, y q “
0. Therefore, we have reduced matters to anincidence problem. We observe that clearly | Π | “ | S | and f p,q ‰ f p ,q if p p, q q ‰ p p , q q , so | Γ | « | P | . Wesummarize these observations in the following lemma: Lemma 1. If S, P, Π , Γ are as above, then | Π | “ | S | , | Γ | « | P | , and Q p S, P q « | S || P | ` I p Π , Γ q . So, matters have been reduced to obtaining an incidence bound that applies to our sets Π and Γ.The incidence bound we will use is one of the versions of a family of Theorems referred to as Pach-Sharir, who usedrestrictions in the incidence graph to generalise Szemer´edi-Trotter. Versions of this result can be found in [4] and[16]. The version we state below as Theorem 3 was reproved using the polynomial method in [11, Theorem 4.1].We first provide the necessary definitions for the Theorem. This requires a definition from algebraic geometry weintroduce the minimum required for our result, for a further introduction see [15, Section 2] and [12, Section 5.3].
Definition 2.
We say a real-algebraic variety f is a real-algebraic curve if the smallest complex algebraic varietythat contains F has dimension one. Definition 3.
The incidence graph on p Π , Γ q is the bipartite graph with vertices Π and Γ and an edge between p u, v q P Π and f p.q P Γ if and only if f p,q p u, v q “ Theorem 3.
Let Γ be a set of real-algebraic curves in R and Π a set of points in R . Suppose that there is no K s,t in the incidence graph on Π , Γ, then I p Π , Γ q À | Π | s s ´ | Γ | s ´ s ´ ` | Π | ` | Γ | . For example if Π is a point set and Γ a line set both in R then the incidence graph doesn’t contain any K , subgraph. In Theorem 3 and Corollary 1 below the bounds improve as s decreases and are independent (up toconstants) of t .We note that our varieties f p,q defined in Section 2 are real-algebraic but they are not real-algebraic curves. Wesolve this problem by restricting f p,q to S ˆ S which as we demonstrate in Section 4 ensures that we are workingwith real-algebraic curves. As our points Π lie in S ˆ S this will not affect either the number of incidences northe K , restriction we prove in the incidence graph.In Section 3 we show for our specific curves that we can take s “ t “
17, however the curves we have will bein R , so to deal with this we will do this using the generic-projection idea used by Solymosi and Tao [20] whichallows us to say that if our curves are in too high an ambient dimension then we can project them down to R in away that allows us to apply an incidence bound in R .This was also done by Pach and de Zuuew [15], so we will quote their result here. Corollary 1. (Corollary 2.4 [15]) Let Γ be a set of real-algebraic curves in R d each defined by e polynomials ofdegree at most D and Π a set of points in R d . Suppose that there is no K s,t in the incidence graph on Π , Γ, then I p Π , Γ q À | Π | s s ´ | Γ | s ´ s ´ ` | Π | ` | Γ | . This was already done for the K , case of Pach-Sharir by Pach and de Zeeuw in [15]. We note that their projectionwill not only preserve the lack of a K , subgraph, but also a K s,t subgraph for any finite s and t .4 Showing that the incidence graph has no K , , Γ contains no K , . The goal is toapply a generalized version of B´ezout’s theorem. A naive application would tell us that if we have 4 polynomialequations of degree 2 in 4 variables, then we must have at most 16 common zeroes. However, this naive extensionof B´ezout is not true, and there are some technical algebraic subtleties that prevent us from proving that there isno K , without putting unreasonable constraints on Π (and hence, on P ).However, adding one more equation (that is, making s “ Lemma 2.
Let p, q P P be distinct and let tp u j , v j q P Π : 1 ď j ď u be five distinct points of Π which are allincident to f p,q . Then, we may choose four of these points which are affinely independent. Proof.
We claim the plane spanned by tp u , v q´p u , v q , p u , v q´p u , v qu can contain only one of p u , v q´p u , v q or p u , v q ´ p u , v q , from which it follows that the points corresponding to either j P t , , , u or j P t , , , u are affinely independent. To prove the claim, if p u j , v j q ´ p u , v q is in that plane, then p u j , v j q ´ p u , v q “ x pp u , v q ´ p u , v qq ` y pp u , v q ´ p u , v qq for some unique values of x and y . Equivalently, we have u j “ x p u ´ u q ` y p u ´ u q ` u v j “ x p v ´ v q ` y p v ´ v q ` v Since } u j } “ } v j } “
1, this means x and y satisfy } x p u ´ u q ` y p u ´ u q ` u } “ } x p v ´ v q ` y p v ´ v q ` v } “ . This system has solutions p x, y q “ p , q , p , q , p , q corresponding to p u , v q , p u , v q , p u , v q , respectively. Toprove the lemma, it therefore suffices to show this system has at most four solutions. For now we assume that bothequations are irreducible quadratics, dealing with the case that these are lines or the product of lines later. If theyare irreducible quadratics, by B´ezout it suffices to prove that one is not a constant multiple of the other. Expandingeach equation and focusing on the quadratic terms while ignoring the lower order terms, we have } u ´ u } x ` } u ´ u } y ` x u ´ u , u ´ u y xy ` ¨ ¨ ¨ “ } v ´ v } x ` } v ´ v } y ` x v ´ v , v ´ v y xy ` ¨ ¨ ¨ “ . (2)Suppose for contradiction that one equation is a constant multiple of the other. This means once we normalize bothequations so that the coefficient of x is 1, and all other coefficients must be the same. In particular, this meanswe have } u ´ u }} u ´ u } “ } v ´ v }} v ´ v } ; denote this common value by t . Let A “ } u ´ u } , B “ } v ´ v } , define θ u to be theangle between u ´ u and u ´ u , and define θ v similarly. Then our equations are A x ` t A y ` p tA cos θ u q xy ` ¨ ¨ ¨ “ B x ` t B y ` p tB cos θ v q xy ` ¨ ¨ ¨ “ x coefficient is 1, x ` t y ` p t cos θ u q xy ` ¨ ¨ ¨ “ x ` t y ` p t cos θ v q xy ` ¨ ¨ ¨ “ . Comparing the xy coefficients, we conclude that θ u “ ˘ θ v . This means that the two triangles ∆ u u u and ∆ v v v are similar; they have common angle θ at the third vertex and side lengths of the form ℓ, tℓ from the third vertexto the first and second, respectively, for some value of ℓ ( ℓ “ A for the first triangle, ℓ “ B for the second).We claim there is only one value of ℓ for which such a triangle has all its vertices on the unit circle. If the claim wetrue it implies } p } “ } q } , as the triangles ∆ u u p and ∆ v v q would be congruent, and this is our contradiction.So, it suffices to prove the claim.Let O denote the origin, and let ∆ αβγ be any triangle with angle θ at α and side lengths αβ “ ℓ, αγ “ tℓ (seefigure 1). O β γ αℓ tℓ θ Figure 1If α, β, γ are on the unit circle then the triangle ∆
Oαβ is an isosceles triangle with common side length 1 andbase length ℓ , hence it has common base angle = Oαβ “ arccos p ℓ { q . This implies = Oαγ “ θ ` arccos p ℓ { q .Similarly, ∆ Oαγ is an isosceles triangle with common side length 1 and base length tℓ , so the common base angleis = Oαγ “ arccos p tℓ { q . This means ℓ must satisfy θ ` arccos ℓ “ arccos tℓ ℓ θ ´ ˆ ´ ℓ ˙ { sin θ “ tℓ ˆ cos θ ´ t ˙ ℓ “ sin θ ˆ ´ ℓ ˙˜ˆ cos θ ´ t ˙ ` sin θ ¸ ℓ “ sin θ. Note the right hand side cannot be zero, since that would imply three points on a circle were also on a line. So,there is at most one positive solution for ℓ .We now show that equations such as (1) are irreducible quadratics. Looking at the full equation we have,6 u ´ u } x ` } u ´ u } y ` x u ´ u , u ´ u y xy ` x u ´ u , u y x ` x u ´ u , u y y “ x and y coefficients being zero mean that all the coefficients are zero. So if (1)is reducible it must be the product of lines. Noting that there is no constant term and normalising so the coefficientof x is 1, we can write these lines as p x ` ay qp x ` by ` c q . Using the same reduction as above we have ab “ t and a ` b “ t cos p θ q . Plugging the first of these into the second gives the quadratic1 a t ´ p θ q t ` a “ , which has a real solution only when cos p θ q “ ˘
1. As noted above, this would mean that we have three points of acircle on a line, a contradiction. So we are not in the case where are quadratics are reducible and thus we are doneby the above.Now that we have this lemma, we can prove that our incidence graph does not contain a K , . An incidenceoccurs between p u, v q P Π and f p,q P Γ if and only if } p ´ u } “ } q ´ v } , so our goal is to prove that for distinct p u j , v j q P Π , ď j ď } p ´ u j } “ } q ´ v j } has at most 16 solutions p, q P P . To do this, we need a version of Bezout’s theorem in dimensions higher than 2. Definition 4. ([1, Disscussion prior to Lemma 11.5.1]) Let P , ..., P n be homogeneous polynomials in variables X , ..., X n . We say x “ p x , ..., x n q is a nonsingular projective solution if P i p x q “ i , and the matrix ˆ B P j B X i p x q ˙ ď j ď n ď i ď n has rank n . Theorem 4 ([1], Lemma 11.5.1) . Let P , ..., P n be homogeneous polynomials in variables X , ..., X n of degrees d , ..., d n , respectively. The number of nonsingular projective solutions is ď ś i d i . Proposition 1.
The incidence graph on p Π , Γ q does not contain a K , . More precisely, let tp u j , v j q P Π : 1 ď j ď u be five distinct points of Π. There are at most 16 pairs p p, q q P P such that } p ´ u j } “ } q ´ v j } . for all j “ , , , , Proof.
By Lemma 2 we can assume without loss of generality that the first four points tp u j , v j qu j “ are affinelyindependent. Let p u j , v j q “ p a j , a j , b j , b j q , consider the four polynomials given by P j p X , ..., X q “ p X ´ a j X q ` p X ´ a j X q ´ p X ´ b j X q ´ p X ´ b j X q for j P t , , , u . These are homogeneous and degree 2, so we will be done if we prove that each zero of the form p p , p , q , q , q with p “ p p , p q , q “ p q , q q P P is non-singular, as we will have shown that these 4 equationshave at most 16 solutions, and hence adding the 5th equation for j “ B P j B X i p p , p , q , q , q ˙ ď j ď ď i ď has rank 4. The i “ ¨˚˚˝ p ´ u q ´ v p ´ u q ´ v p ´ u q ´ v p ´ u q ´ v ˛‹‹‚ has rank 4. Note that each entry is viewed as a 1 ˆ ˆ “ ÿ j “ α j p p ´ u j , q ´ v j q , where not all the α j are zero. Let A “ ř j “ α j . Then we have0 “ ÿ j “ α j p p ´ u j , q ´ v j q “ A p p, q q ´ ÿ j “ α j p u j , v j q , or A p p, q q “ ÿ j “ α j p u j , v j q . Furthermore, we have that A ‰ tp u j , v j q : 1 ď j ď u is affinely independent. Thus, the above gives us p “ ÿ j “ α j A u j ,q “ ÿ j “ α j A v j . Also, expanding out the equation } p ´ u j } “ } q ´ v j } in terms of inner products and using the fact that u j , v j are on the unit circle gives } p } ´ x p, u j y “ } q } ´ x q, v j y . Recall A “ ř j α j , thus multiplying both sides by ´ α j { A and summing, the left hand side is } p } and the righthand side is } q } . This contradicts our assumption that P has no two points on a concentric circle. Thus, thismatrix is full rank, and we are done. 8 Applying the incidence bound
We wish to use Corollary 1 on our varieties f p,q in Γ. However, our varieties f p,q are defined by one equation, seeEquation (2), in R , so have no hope at this stage of being real-algebraic curves. We get around this by limitingour curves to S ˆ S , the domain of our point set Π. We define new varieties C p,q as C p,q “ p S ˆ S q X Z p f p,q q which we will show are real-algebraic curves. It is clear that there is no K , with our new curves C p,q as these arejust restrictions of f p,q which we showed above cannot give a K , .We will prove that the C p,q are real-algebraic curve by following the technique of Pach and de Zeeuw [15, Lemma3.3] demonstrating that C p.q are curves of complex-dimension one. We will then apply Corollary 1 to these curves.Before we prove this we state the following theorem from algebraic geometry. It can be found in many standardreferences such as [10, Exercise 11.6], [18, Chapter 1, Section 6, Theorem 4]. Theorem 5. If X is an irreducible variety in C d and F any polynomial in C r x , . . . , x d s that does not vanish onthe whole of X then dim p X X Z p F qq ď dim p X q ´ C p,q are real-algebraic curves we require some routine results from algebraic geometry, these results requireus to work over an algebraically closed field. Thus we introduce the complexification of a real variety V . Definition 5.
Suppose V is a real-algebraic variety. Let V C be the smallest complex variety which contains V , wenote that such a variety always exists and the V is its set of real points [22]. We call V C the complexification of V . Lemma 3. C C p,q has complex dimension at most one, and thus C p,q are real-algebraic curves or a finite union ofpoints. Proof.
We note that p S q C is irreducible and of complex dimension one. From [10, Exercise 5.9] the product of twoirreducible varieties is irreducible, so we have that p S q C ˆ p S q C “ p S ˆ S q C is irreducible. We then use [10,discussion prior to Theorem 11.12] to say that p S ˆ S q C has complex-dimension two.We now show that f C p,q cannot vanish identically on p S ˆ S q C , so that we can apply Theorem 5. To do this wenote that f C p,q p u, v q “ } u ´ p } “ } v ´ q } . If this is true for all points u and v in S (the real version suffices here) then it means that p and q are equidistantfrom every point on the circle. But this means that p “ q “
0, the centre of the circle. However we have alreadydealt with the case p “ q in Section 2, so this is a contradiction. Thus f C p,q cannot vanish on the whole of p S ˆ S q C and by Theorem 5 we get the C p,q “ pp S ˆ S q X Z p f p,q qq has complex-dimension at most one, as claimed.We note that if C C p,q has dimension smaller than one it is a finite union of points.We use a bound due to Oleinik-Petrovskii [17], Milnor [14] and Thom [21] which bounds the number of connectedcomponents of a real-algebraic variety. Theorem 6.
An algebraic variety in R d definied by polynomials of degree at most D has at most p D q d connectedcomponents.As we have d “ D “ C | Γ | “ C | P | which will suffice.Now we consider the case where all of our curves C C p,q have dimension one and thus all our C p,q are real-algebraiccurves. We can now apply Corollary 1 to our curves r Γ “ t C p,q u p,q P P and our points Π. I p Π , r Γ q À | Π | | r Γ | ` | Π | ` | r Γ | . Using Lemma 1 together with the fact that I p Π , Γ q “ I p Π , r Γ q , we have | Q p S, P q| « | S || P | ` I p Π , Γ q À | S | | P | ` | S | ` | P | . | Q p S, P q|| ∆ p S, P q| ě | S | | P | which when we apply it shows that | ∆ p S, P q| Á min t| S | | P | , | S | , | P | u as claimed. References [1] Jacek Bochnak, Michel Coste, and Marie-Fran¸coise Roy.
Real algebraic geometry , volume 36. Springer Science& Business Media, 2013. 7[2] Peter Brass, William OJ Moser, and J´anos Pach.
Research problems in discrete geometry . Springer Science &Business Media, 2006. 1[3] Ariel Bruner and Micha Sharir. Distinct distances between a collinear set and an arbitrary set of points.
Discrete Mathematics , 341(1):261–265, 2018. 1, 2, 3[4] Kenneth L Clarkson, Herbert Edelsbrunner, Leonidas J Guibas, Micha Sharir, and Emo Welzl. Combinatorialcomplexity bounds for arrangements of curves and spheres.
Discrete & Computational Geometry , 5(2):99–160,1990. 4[5] Gy¨orgy Elekes. A note on the number of distinct distances.
Periodica Mathematica Hungarica , 38(3):173–177,1999. 1, 2[6] Gy¨orgy Elekes and Lajos R´onyai. A combinatorial problem on polynomials and rational functions.
Journal ofCombinatorial Theory, Series A , 89(1):1–20, 2000. 1[7] Paul Erd˝os. On sets of distances of n points.
The American Mathematical Monthly , 77(7):738–740, 1970. 1[8] Julia Garibaldi, Alex Iosevich, and Steven Senger.
The Erdos distance problem , volume 56. American Mathe-matical Soc., 2011. 1[9] Larry Guth and Nets Hawk Katz. On the erd˝os distinct distances problem in the plane.
Annals of mathematics ,pages 155–190, 2015. 1[10] Joe Harris.
Algebraic geometry: a first course , volume 133. Springer Science & Business Media, 2013. 9[11] Haim Kaplan, Jiˇr´ı Matouˇsek, and Micha Sharir. Simple proofs of classical theorems in discrete geometry viathe guth–katz polynomial partitioning technique.
Discrete & Computational Geometry , 48(3):499–517, 2012. 4[12] Surya Mathialagan. On bipartite distinct distances in the plane. arXiv preprint arXiv:1912.01883 , 2019. 1, 2,3, 4[13] Bryce McLaughlin and Mohamed Omar. On distinct distances between a variety and a point set. arXiv preprintarXiv:1812.03371 , 2018. 1, 2[14] John Milnor. On the betti numbers of real varieties.
Proceedings of the American Mathematical Society ,15(2):275–280, 1964. 9[15] J´anos Pach and Frank De Zeeuw. Distinct distances on algebraic curves in the plane.
Combinatorics, Probabilityand Computing , 26(1):99–117, 2017. 1, 2, 3, 4, 9[16] J´anos Pach and Micha Sharir. On the number of incidences between points and curves.
Combinatorics,Probability and Computing , 7(1):121–127, 1998. 4[17] Ivan Georgievich Petrovskii and Olga Arsen’evna Oleinik. On the topology of real algebraic surfaces.
IzvestiyaRossiiskoi Akademii Nauk. Seriya Matematicheskaya , 13(5):389–402, 1949. 9[18] Igor R Shafarevich. Basic algebraic geometry. grundlehren 213, 1974. 91019] Micha Sharir, Adam Sheffer, and J´ozsef Solymosi. Distinct distances on two lines.
Journal of CombinatorialTheory, Series A , 120(7):1732–1736, 2013. 1, 2[20] J´ozsef Solymosi and Terence Tao. An incidence theorem in higher dimensions.
Discrete & ComputationalGeometry , 48(2):255–280, 2012. 4[21] Ren´e Thom. Sur l’homologie des vari´et´es alg´ebriques r´eelles. In
Differential and Combinatorial Topology (ASymposium in Honor of Marston Morse) , pages 255–265. Princeton Univ. Press, Princeton, N.J., 1965. 9[22] Hassler Whitney. Elementary structure of real algebraic varieties.
Annals of Mathematics , pages 545–556,1957. 9
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